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Homological stability for spaces of embedded surfaces

Federico Cantero Morán and Oscar Randal-Williams

Geometry & Topology 21 (2017) 1387–1467

We study the space of oriented genus-g subsurfaces of a fixed manifold M and, in particular, its homological properties. We construct a “scanning map” which compares this space to the space of sections of a certain fibre bundle over M associated to its tangent bundle, and show that this map induces an isomorphism on homology in a range of degrees.

Our results are analogous to McDuff’s theorem on configuration spaces, extended from 0–dimensional submanifolds to 2–dimensional submanifolds.

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submanifolds, characteristic classes, homology stability, embedding spaces, mapping class groups, scanning
Mathematical Subject Classification 2010
Primary: 55R40, 57R20, 57R40, 57R50, 57S05
Received: 14 July 2014
Revised: 14 March 2016
Accepted: 29 May 2016
Published: 10 May 2017
Proposed: Shigeyuki Morita
Seconded: Peter Teichner, Benson Farb
Federico Cantero Morán
Institut de Recherche en Mathématique et Physique
Université Catholique de Louvain
Chemin du Cyclotron, 2
1348 Louvain-la-Neuve
Oscar Randal-Williams
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB
United Kingdom